In this chapter, we explore several important statistical models. Statistical models allow us to perform statistical inference—the process of selecting models and making predictions about the underlying distributions—based on the data we have. Many approaches exist, from the stochastic block model and its generalizations to the edge observer model, the exponential random graph model, and the graphical LASSO. As we show in this chapter, such models help us understand our data, but using them may at times be challenging, either computationally or mathematically. For example, the model must often be specified with great care, lest it seize on a drastically unexpected network property or fall victim to degeneracy. Or the model must make implausibly strong assumptions, such as conditionally independent edges, leading us to question its applicability to our problem. Or even our data may be too large for the inference method to handle efficiently. As we discuss, the search continues for better, more tractable statistical models and more efficient, more accurate inference algorithms for network data.

This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.

This chapter examines discrete-time LTI systems in detail. It shows that the input–output behavior of an LTI system is characterized by the so-called impulse response. The output is shown to be the so-called convolution of the input with the impulse response. It is then shown that exponentials are eigenfunctions of LTI systems. This property leads to the ideas of transfer functions and frequency responses for LTI systems. It is argued that the frequency response gives a systematic meaning to the term “filtering.” Image filtering is demonstrated with examples. The discrete-time Fourier transform (DTFT) is introduced to describe the frequency domain behavior of LTI systems, and allows one to represent a signal as a superposition of single-frequency signals (the Fourier representation). DTFT is discussed in detail, with many examples. The z-transform, which is of great importance in the study of LTI systems, is also introduced and its connection to the Fourier transform explained. Attention is also given to real signals and real filters, because of their additional properties in the frequency domain. Homogeneous time-invariant (HTI) systems are also introduced. Continuous-time counterparts of these topics are explained. B-splines, which arise as examples in continuous-time convolution, are presented.

This chapter discusses many interesting properties of bandlimited signals. The subspace of bandlimited signals is introduced. It is shown that uniformly shifted versions of an appropriately chosen sinc function constitute an orthogonal basis for this subspace. It is also shown that the integral and the energy of a bandlimited signal can be obtained exactly from samples if the sampling rate is high enough. For non-bandlimited functions, such a result is only approximately true, with the approximation getting better as the sampling rate increases. A number of less obvious consequences of these results are also presented. Thus, well-known mathematical identities are derived using sampling theory. For example, the Madhava–Leibniz formula for the approximation of π can be derived like this. When samples of a bandlimited signal are contaminated with noise, the reconstructed signal is also noisy. This noise depends on the reconstruction filter, which in general is not unique. Excess bandwidth in this filter increases the noise, and this is quantitatively analyzed. An interesting connection between bandlimited signals and analytic functions (entire functions) is then presented. This has many implications, one being that bandlimited signals are infinitely smooth.

In working with network data, data acquisition is often the most basic yet the most important and challenging step. The availability of data and norms around data vary drastically across different areas and types of research. A team of biologists may spend more than a decade running assays to gather a cells interactome; another team of biologists may only analyze publicly available data. A social scientist may spend years conducting surveys of underrepresented groups. A computational social scientist may examine the entire network of Facebook. An economist may comb through large financial documents to gather tables of data on stakes in corporate holdings. In this chapter, we move one step along the network study life-cycle. Key to data gathering is good record-keeping and data provenance. Good data gathering sets us up for future success—otherwise, garbage in, garbage out—making it critical to ensure the best quality and most appropriate data is used to power your investigation.

This chapter covers ways to explore your network data using visual means and basic summary statistics, and how to apply statistical models to validate aspects of the data. Data analysis can generally be divided into two main approaches, exploratory and confirmatory. Exploratory data analysis (EDA) is a pillar of statistics and data mining and we can leverage existing techniques when working with networks. However, we can also use specialized techniques for network data and uncover insights that general-purpose EDA tools, which neglect the network nature of our data, may miss. Confirmatory analysis, on the other hand, grounds the researcher with specific, preexisting hypotheses or theories, and then seeks to understand whether the given data either support or refute the preexisting knowledge. Thus, complementing EDA, we can define statistical models for properties of the network, such as the degree distribution, or for the network structure itself. Fitting and analyzing these models then recapitulates effectively all of statistical inference, including hypothesis testing and Bayesian inference.

This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.

Realistic networks are rich in information. Often too rich for all that information to be easily conveyed. Summarizing the network then becomes useful, often necessary, for communication and understanding but, being wary, of course, that a summary necessarily loses information about the network. Further, networks often do not exist in isolation. Multiple networks may arise from a given dataset or multiple datasets may each give rise to different views of the same network. In such cases and more, researchers need tools and techniques to compare and contrast those networks. In this chapter, In this chapter, well show you how to summarize a network, using statistics, visualizations, and even other networks. From these summaries we then describe ways to compare networks, defining a distance between networks for example. Comparing multiple networks using the techniques we describe can help researchers choose the best data processing options and unearth intriguing similarities and differences between networks in diverse fields.

This chapter introduces the discrete Fourier transform (DFT), which is different from the discrete-time Fourier transform (DTFT) introduced earlier. The DFT transforms an N-point sequence x[n] in the time domain to an N-point sequence X[k] in the frequency domain by sampling the DTFT of x[n]. A matrix representation for this transformation is introduced, and the properties of the DFT matrix are studied. The fast Fourier transform (FFT), which is a fast algorithm to compute the DFT, is also introduced. The FFT makes the computation of the Fourier transforms of large sets of data practical. The digital signal processing revolution of the 1960s was possible because of the FFT. This chapter introduces the simplest form of FFT, called the radix-2 FFT, and a number of its properties. The chapter also introduces circular or cyclic convolution, which has a special place in DFT theory, and explains the connection to ordinary convolution. Circular convolution paves the way for fast algorithms for ordinary convolution, using the FFT. The chapter also summarizes the relationships between the four types of Fourier transform studied in this book: CTFT, DTFT, DFT, and Fourier series.

Machine learning, especially neural network methods, is increasingly important in network analysis. This chapter will discuss the theoretical aspects of network embedding methods and graph neural networks. As we have seen, much of the success of advanced machine learning is thanks to useful representations—embeddings—of data. Embedding and machine learning are closely aligned. Translating network elements to embedding vectors and sending those vectors as features to a predictive model often leads to a simpler, more performant model than trying to work directly with the network. Embeddings help with network learning tasks, from node classification to link prediction. We can even embed entire networks and then use models to summarize and compare networks. But not only does machine learning benefit from embeddings, but embeddings benefit from machine learning. Inspired by the incredible recent progress with natural language data, embeddings created by predictive models are becoming more useful and important. Often these embeddings are produced by neural networks of various flavors, and we explore current approaches for using neural networks on network data.

This chapter discusses record keeping, like maintaining a lab notebook. Historically, lab notebooks were analog, pen-and-paper affairs. With so much work being performed on the computer and with most scientific instruments creating digital data directly, most record-keeping efforts are digital. Therefore, we focus on strategies for establishing and maintaining records of computer-based work. Keeping good records of your work is essential. These records inform your future thoughts as you reflect on the work you have already done, acting as reminders and inspiration. They also provide important details for collaborators, and scientists working in large groups often have predefined standards for group members to use when keeping lab notebooks and the like. Computational work differs from traditional bench science, and this chapter describes practices for good record-keeping habits in the more slippery world of computer work.